Ph.D. COMPREHENSIVE EXAM REAL ANALYSIS January 1994

Justify your answers. Four complete solutions will constitute a passing score.

1. Let A be a σ-algebra of subsets of a set X.

a. Suppose that f is a function from X into the extended real numbers. Define

the phrase f is measurable. b. Suppose that f : ℝ → ℝ is differentiable at each point of ℝ. Prove that f′ is

Borel measurable.

2. With the same notation as in Problem 1, define a set A ∈ A to be an atom of a measure μ on (X, A) if μ(A) > 0, and for all B ⊂ A such that B ∈ A, we have μ(B) = μ(A) or μ(B) = 0. a. Suppose that f is a right-continuous non-decreasing real-valued function on

[0, 1]. Prove that x_{0} is a discontinuity of f if and only if {x_{0}} is an atom of the induced measure μ_{f}. (μ_{f}( (a, b] ) = f(b) − f(a), for all a, b ∈ [0, 1].)

b. Give an example of a strictly increasing function g on [0,1] such that g′ = 0

almost everywhere with respect to Lebesgue measure. c. Give an example (sketch the construction and verification) of a non-decreasing

function h on [0,1] such that h′ = 0 almost everywhere with respect to Lebesgue measure but the induced measure μ_{h} has no atoms.

3. a. Define the term Hilbert space. (Assume the scalar field is ℂ.)

b. Define orthonormal system and complete orthonormal system in a Hilbert

space H. c. Let ρ_{n}(t) = signum(sin(2^{n}πt)) for t ∈ [0, 1] and n = 0, 1, 2, …. Show

that (ρ_{n})_{n} is an orthonormal system in L^{2}[0, 1] (with Lebesgue measure). ( signum(x) = x/|x| if x ≠ 0 and signum(x) = 0 if x = 0.)

d. Prove that for all f ∈ L^{2}[0, 1], lim_{n→∞} ∫ fρ_{n} = 0.

4. a. Suppose that K is a compact metric space, state the Riesz Representation Theorem for C(K)^{∗}. b. Prove that for all f ∈ C[0, 1],

lim_{n→∞} ∑_{k=1}^{n} \frac{k}{n^{2}} f(\frac{k}{n}) exists.

c. Prove that there is a Borel measure μ on [0,1] such that

lim_{n→∞} ∑_{k=1}^{n} \frac{k}{n^{2}} f(\frac{k}{n}) = ∫ f dμ

for all f ∈ C[0, 1].

5. a. Suppose that f is a continuous real-valued function on (0,1], f ≥ 0 and the improper Riemann integral of f on (0,1] is finite. Prove that f is Lebesgue integrable on [0,1]. b. Prove or disprove: The assumption f ≥ 0 is necessary.

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